%% term_grad_Pold_V
function vector_b = term_grad_Pold_V(mesh, fespace1, fespace2, Pold)
Nb1 = fespace1.Nb;
Nlb1 = fespace1.Nlb;
rows = zeros(mesh.Ne*Nlb1,1);
val1 = rows;
val2 = rows;
flag = 1;
for n = 1:mesh.Ne
    V = mesh.node(mesh.elem(n,:),:);
    [int_points_loc, int_weights_loc] = integral_loc(V, "2d");
    phi = basis_loc(V, int_points_loc(:,1), int_points_loc(:,2), fespace1.basis_type, "x");
    psix = basis_loc(V, int_points_loc(:,1), int_points_loc(:,2), fespace2.basis_type, "dx");
    psiy = basis_loc(V, int_points_loc(:,1), int_points_loc(:,2), fespace2.basis_type, "dy");
    fp1 = psix * Pold(fespace2.elem(n,:));
    fp2 = psiy * Pold(fespace2.elem(n,:));
    for i = 1:Nlb1
        f1 = fp1 .* phi(:,i);
        f2 = fp2 .* phi(:,i);
        r1 = int_weights_loc * f1;
        r2 = int_weights_loc * f2;
        rows(flag) = fespace1.elem(n,i);
        val1(flag) = r1;
        val2(flag) = r2;
        flag = flag + 1;
    end
end
b1 = sparse(rows, 1, val1, Nb1, 1);
b2 = sparse(rows, 1, val2, Nb1, 1);
vector_b = [b1; b2];
end